Sufficient Conditions For Asymptotic Normality Research Articles

Asymptotic normality of winsorized means

Let X i be non-degenerate i.i.d. random variables with distribution function F, and let X n1 ,…, X nn denote the order statistics of X 1,…, X n . In trying to robustify the sample mean as an estimator of location, several alternatives have been suggested which have the intuitive appeal of being less susceptible to outliers. Here the asymptotic distribution of one of these, the Winsorized mean, which is given by n -1 S nX n,s n + ∑ i=s n+1 n−r nX ni+r nX n,n−r n+1 where r n ⩾0, s n ⩾0 and r n + s n ⩾ n, is studied. The main results include a necessary and sufficient condition for asymptotic normality of the Winsorized mean under the assumption that r n →∞, s n →∞, r n n −1→0, s n n −1→0 and F is convex at infinity. It is also shown, perhaps somewhat surprisingly, that if the convexity assumption on F is dropped then the Winsorized mean may fail to be asymptotically normal even when X 1 is bounded!

On the exponential model for survival

SUMMARY A simple sufficient condition for the existence, consistency and asymptotic normality of the regression coefficient in the exponential model for survival is given. The intuition that consistent estimation can only be guaranteed when censoring variables are not too small, depending on the values of the covariates, can be quantified in some examples with the use of this condition. The exponential model for survival time is a fundamental building block in survival analysis, yet there seems to be no easily accessible condition which guarantees existence, consistency and asymptotic normality of the estimators. Well-known heuristics, such as those of Kalbfleisch & Prentice (1980, p. 51), suggest that a sufficient condition will be uniform asymptotic negligibility of the sequence of covariate vectors by comparison with the information matrix. We show that this is indeed the case; see Condition 1 below. The simplicity of this condition encourages applications, and we demonstrate, for example, how Condition 1 reveals the role of censoring in the consistency of the estimator. Since an analogue of Condition 1 is a necessary and sufficient condition for asymptotic normality of the regression coefficients in the ordinary least-squares model (Eicker, 1965), it is probably as general as could be hoped for. Our method is to extend to the survival context the methods of Fahrmeir & Kaufmann (1985), who, in the area of generalized linear models, took advantage of the 'close to linear' structure of those models to strengthen traditional asymptotic arguments for general maximum likelihood estimators. The asymptotics of survival models have been widely studied using the theory of counting processes and martingales, and Borgan (1984) uses this theory to deal with a wide class of models with covariates. To apply his results (Borgan, 1984, p. 14), however, very technical conditions involving uniform convergence of various stochastic processes must be verified. Borgan's methods rely on a traditional approach and cannot yield simple sufficient conditions like Condition 1. We need nothing more sophisticated than Chebychev's inequality and the central limit theorem for sums of independent random variables, although we are restricted to the 'close to linear' exponential model. Another approach by Friedman (1982) requires complex assumptions

Central limit theorem for quadratic forms for sparse tables

Sufficient conditions for asymptotic normality for quadratic forms in { n t − np t } are given, where { n t } are the observed counts with expected cell means { np t }. The main result is used to derive asymptotic distributions of many statistics including the Pearson's chi-square.

Successive Sampling in Large Finite Populations

The permutation distribution induced upon a finite population by the order of selection under successive sampling is closely related to the order statistics of independent exponentially distributed waiting times. This characterization is applied to obtain necessary and sufficient conditions for asymptotic normality of the sum of characteristics observed in a successive sample from a finite population. The necessary and sufficient conditions generalize previous results for simple random sampling without replacement, and apply to sampling fractions close to 0 or 1.

Asymptotic Normality of Permutation Statistics Derived from Weighted Sums of Bivariate Functions

Statistics of the form $H_n = \sum d_{ijn}h_n(X_i, X_j)$ are considered, where $X_1, X_2, \cdots$, are independent and identically distributed random variables, the diagonal terms, $d_{iin}$, are equal to zero, and $h_n(x, y)$ is a symmetric real valued function. The asymptotic normality of such statistics is proven and the result then combined with work of Jogdeo on statistics that are weighted sums of bivariate functions of ranks to find sufficient conditions for asymptotic normality of permutation statistics derived from $H_n$.

Sufficient conditions for asymptotic normality of sums of values of discrete random fields, dependent in strips

Sufficient conditions are given for the asymptotic normality of a sum of values of a field on a square of the integer lattice in R2 with the side n → ∞ for normalization by the quantity (2n+1)3/2. Moreover, the values of the field are dependent on strips of fixed width. The method of moments is used for proof.

Some Properties of Incomplete U-Statistics

SUMMARY Let g be a symmetric function with k arguments. A U-statistic is the arithmetic mean of g's based on the N = n!/{k! (n- k)!} subsamples of size k taken from a sample of size n. When N is large, it may be convenient to use instead an 'incomplete' U-statistic based on m suitably selected subsamples. The variance of such a statistic is studied, exactly and asymptotically for large m and n. It is shown that an incomplete statistic may be asymptotically efficient compared with the 'complete' one even when m increases much less rapidly than N. Some sufficient conditions for asymptotic normality of an incomplete U-statistic are given.

Some properties of incomplete U-statistics

SUMMARY Let g be a symmetric function with k arguments. A U-statistic is the arithmetic mean of g's based on the N = n!/{k! (n- k)!} subsamples of size k taken from a sample of size n. When N is large, it may be convenient to use instead an 'incomplete' U-statistic based on m suitably selected subsamples. The variance of such a statistic is studied, exactly and asymptotically for large m and n. It is shown that an incomplete statistic may be asymptotically efficient compared with the 'complete' one even when m increases much less rapidly than N. Some sufficient conditions for asymptotic normality of an incomplete U-statistic are given.

Asymptotic Theory of Rejective Sampling with Varying Probabilities from a Finite Population

In [3] the author established necessary and sufficient conditions for asymptotic normality of estimates based on simple random sampling without replacement from a finite population, and thus solved a comparatively old problem initiated by W. G. Madow [8]. The solution was obtained by approximating simple random sampling by so called Poisson sampling, which may be decomposed into independent subexperiments, each associated with a single unit in the population. In the present paper the same method is used for deriving asymptotic normality conditions for a special kind of sampling with varying probabilities called here rejective sampling. Rejective sampling may be realized by $n$ independent draws of one unit with fixed probabilities, generally varying from unit to unit, given the condition that samples in which all units are not distinct are rejected. If the drawing probabilities are constant, rejective sampling coincides, with simple random sampling without replacement, and so the present paper is a generalization of [3]. Basic facts about rejective sampling are exposed in Section 2. To obtain more refined results, Poisson sampling is introduced and analyzed (Section 3) and then related to rejective sampling (Section 4). Next three sections deal with probabilities of inclusion, variance formulas and asymptotic normality of estimators for rejective sampling. In Section 8 asymptotic formulas are tested numerically and applications to sample surveys are indicated. The paper is concluded by short-cuts in practical performance of rejective sampling. The readers interested in applications only may concentrate upon Sections 1, 8 and 9. Those interested in the theory of mean values and variances only, may omit Lemma 4.3 and Section 7.

Robust Estimation of a Location Parameter

Robust Estimation of a Location Parameter